A systematic approach to matrix forms of the Pascal triangle: The twelve triangular matrix forms and relations

This work initiates a systematic investigation into the matrix forms of the Pascal triangle as mathematical objects in their own right. The present paper is especially devoted to the so-called G-matrices, i.e. the set of the twelve (n + 1) x (n + 1) triangular matrix forms that can be derived from the Pascal triangle expanded to the level n (2 <= n is an element of N). For n = 1, the G-matrix set reduces to a set of four distinct matrices. The twelve G-matrices are defined and the classic Pascal recursion is reformulated for each of the twelve G-matrices. Three sets of matrix transformations are then introduced to highlight different relations between the twelve G-matrices and for generating them from appropriately chosen subsets. (C) 2009 Elsevier Ltd. All rights reserved.